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<meta name="description" content="支持向量机(Support Vector Machine, SVM)是一种基于统计学习的模式识别的分类方法，主要用于模式识别。所谓支持向量指的是在分割区域边缘的训练样本点，机是指算法。就是要找到具有最大间隔的分隔面。实际上解决的是一个最优分类器设计的问题。问题目的:找到一个最优分类器，即找到一个分类器，使得分类间隔最大。优化的目标函数:分类间隔，需要使得分类间隔最大。优化对象:分类超平面(决策平面">
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          <h1 class="post-title" itemprop="name headline">量化投资学习笔记22——回归分析:支持向量机</h1>
        

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        <p>支持向量机(Support Vector Machine, SVM)是一种基于统计学习的模式识别的分类方法，主要用于模式识别。所谓支持向量指的是在分割区域边缘的训练样本点，机是指算法。就是要找到具有最大间隔的分隔面。实际上解决的是一个最优分类器设计的问题。<br>问题<br>目的:找到一个最优分类器，即找到一个分类器，使得分类间隔最大。<br>优化的目标函数:分类间隔，需要使得分类间隔最大。<br>优化对象:分类超平面(决策平面)，通过调整分类超平面的位置，使得间隔最大，实现优化目标。<br><img src="https://zymblog-1258069789.cos.ap-chengdu.myqcloud.com/blog0178-QTLearn/15/01.png"><br>超平面(Hyperplane)，指n维欧氏空间中余维度等于1的线性子空间。二维空间中为一条直线，三维空间中为一个二维平面。<br>间隔:支持向量对应点到分类超平面的垂直距离的两倍。即W =2d。<br>现在要做的是，在所有的样本点中，找到合适的支持向量，在保证分类正确的前提下，让间隔W = 2d最大。<br>再往后就是具体的求解推导的过程了，听听就行了。<br>对于线性不可分的情况，考虑将样本映射到更高维的空间中去，希望在这个高维空间中其线性可分。<br>例:一条直线上的两个不同分类的点也许不可分，将其映射到二维平面里也许就可以区分了。<br><img src="https://zymblog-1258069789.cos.ap-chengdu.myqcloud.com/blog0178-QTLearn/15/02.png"><br>如果原始空间是有限维，即属性数有限，一定存在一个高维特征空间使样本线性可分。<br>这就引出了核函数的概念。K(x, x’) = φ(x)·φ(x’)<br>当后者不容易求时，可找到一个函数K，即为核函数。<br>推导看不懂。<br>选择核函数无明确的指导原则，常用RBF，其次是线性核。<br>异常点造成的非线性，SVM允许在一定程度上偏离一下超平面。<br>SVM多分类<br>直接法:将多分类面的参数求解合并到一个最优化问题中。<br>间接法:组合多个二分类SVM分类器<br>有一对一法和一对多法。<br>下面来实践，还是使用iris数据。参考<a target="_blank" rel="noopener" href="https://blog.csdn.net/u012679707/article/details/80501358">https://blog.csdn.net/u012679707/article/details/80501358</a></p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> matplotlib.pyplot <span class="keyword">as</span> plt</span><br><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"><span class="keyword">from</span> sklearn.model_selection <span class="keyword">import</span> train_test_split</span><br><span class="line"><span class="keyword">from</span> sklearn <span class="keyword">import</span> svm</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="comment"># 转换类别</span></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">Iris_label</span>(<span class="params">s</span>):</span></span><br><span class="line">    it = &#123;<span class="string">b&#x27;Iris-setosa&#x27;</span>:<span class="number">0</span>, <span class="string">b&#x27;Iris-versicolor&#x27;</span>:<span class="number">1</span>, <span class="string">b&#x27;Iris-virginica&#x27;</span>:<span class="number">2</span>&#125;</span><br><span class="line">    <span class="keyword">return</span> it[s]</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="keyword">if</span> __name__ == <span class="string">&quot;__main__&quot;</span>:</span><br><span class="line">    <span class="comment"># 读取数据</span></span><br><span class="line">    data = np.loadtxt(<span class="string">&quot;iris.data&quot;</span>, dtype = <span class="built_in">float</span>, delimiter = <span class="string">&#x27;,&#x27;</span>, converters = &#123;<span class="number">4</span> : Iris_label&#125;)</span><br><span class="line">   </span><br><span class="line">    print(data)</span><br><span class="line">   </span><br><span class="line">    <span class="comment"># 划分数据与标签</span></span><br><span class="line">    x, y = np.split(data, indices_or_sections = (<span class="number">4</span>,), axis = <span class="number">1</span>)</span><br><span class="line">    x = x[:, <span class="number">0</span>:<span class="number">2</span>] <span class="comment"># 为了画图，只选前两列</span></span><br><span class="line">    train_data, test_data, train_label, test_label = train_test_split(x, y, random_state = <span class="number">1</span>, train_size = <span class="number">0.6</span>, test_size = <span class="number">0.4</span>)</span><br><span class="line">    print(<span class="string">&quot;训练集大小:&quot;</span>, train_data.shape)</span><br><span class="line">    print(train_data)</span><br><span class="line">    print(test_data)</span><br><span class="line">   </span><br><span class="line">    <span class="comment"># 训练svm分类器</span></span><br><span class="line">    classifier = svm.SVC(C = <span class="number">2</span>, kernel = <span class="string">&quot;rbf&quot;</span>, gamma = <span class="number">10</span>, decision_function_shape = <span class="string">&quot;ovr&quot;</span>) <span class="comment">#ovr 一对多策略</span></span><br><span class="line">    classifier.fit(train_data, train_label.ravel())</span><br><span class="line">   </span><br><span class="line">    <span class="comment"># 计算分类准确率</span></span><br><span class="line">    print(<span class="string">&quot;训练集:&quot;</span>, classifier.score(train_data, train_label))</span><br><span class="line">    print(<span class="string">&quot;测试集:&quot;</span>, classifier.score(test_data, test_label))</span><br><span class="line">   </span><br><span class="line">    <span class="comment"># 查看决策函数</span></span><br><span class="line">    print(<span class="string">&quot;训练决策函数:&quot;</span>, classifier.decision_function(train_data))</span><br><span class="line">    print(<span class="string">&quot;预测结果:&quot;</span>, classifier.predict(train_data))</span><br></pre></td></tr></table></figure>
<p>结果:<br>训练集: 0.8555555555555555<br>测试集: 0.7<br>训练集比测试集结果好。<br>再画图看看。</p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment"># 绘图</span></span><br><span class="line">    fig = plt.figure()</span><br><span class="line">    x1_min, x1_max = x[:, <span class="number">0</span>].<span class="built_in">min</span>(), x[:, <span class="number">0</span>].<span class="built_in">max</span>()</span><br><span class="line">    x2_min, x2_max = x[:, <span class="number">1</span>].<span class="built_in">min</span>(), x[:, <span class="number">1</span>].<span class="built_in">max</span>()</span><br><span class="line">    x1, x2 = np.mgrid[x1_min:x1_max:<span class="number">200j</span>, x2_min:x2_max:<span class="number">200j</span>]</span><br><span class="line">    grid_test = np.stack((x1.flat, x2.flat), axis = <span class="number">1</span>)</span><br><span class="line">   </span><br><span class="line">    <span class="comment"># 设置颜色</span></span><br><span class="line">    cm_light = ListedColormap([<span class="string">&#x27;#A0FFA0&#x27;</span>, <span class="string">&#x27;#FFA0A0&#x27;</span>, <span class="string">&#x27;#A0A0FF&#x27;</span>])</span><br><span class="line">    cm_dark = ListedColormap([<span class="string">&#x27;g&#x27;</span>,<span class="string">&#x27;r&#x27;</span>,<span class="string">&#x27;b&#x27;</span>])</span><br><span class="line">    grid_hat = classifier.predict(grid_test)</span><br><span class="line">    grid_hat = grid_hat.reshape(x1.shape)</span><br><span class="line">    <span class="comment"># 绘图</span></span><br><span class="line">    plt.pcolormesh(x1, x2, grid_hat, cmap = cm_light)</span><br><span class="line">    plt.scatter(x[:, <span class="number">0</span>], x[:, <span class="number">1</span>], c = y[:, <span class="number">0</span>], s = <span class="number">30</span>, cmap = cm_dark)</span><br><span class="line">    plt.scatter(test_data[:, <span class="number">0</span>], test_data[:, <span class="number">1</span>], c = test_label[:, <span class="number">0</span>], s = <span class="number">30</span>, edgecolors = <span class="string">&quot;k&quot;</span>, zorder = <span class="number">2</span>, cmap = cm_dark)</span><br><span class="line">    plt.xlabel(<span class="string">&quot;length&quot;</span>)</span><br><span class="line">    plt.ylabel(<span class="string">&quot;width&quot;</span>)</span><br><span class="line">    plt.xlim(x1_min, x1_max)</span><br><span class="line">    plt.ylim(x2_min, x2_max)</span><br><span class="line">    plt.savefig(<span class="string">&quot;result.png&quot;</span>)</span><br></pre></td></tr></table></figure>
<p><img src="https://zymblog-1258069789.cos.ap-chengdu.myqcloud.com/blog0178-QTLearn/15/03.png"><br>试一下把四列数据都进行建模的结果:<br>四列数据都进行建模的结果<br>训练集: 1.0<br>测试集: 0.95<br>结果很好。<br>接下来用这个方法解决一下泰坦尼克号问题吧。下次。<br>本文代码:<br><a target="_blank" rel="noopener" href="https://github.com/zwdnet/MyQuant/tree/master/21">https://github.com/zwdnet/MyQuant/tree/master/21</a></p>
<p>我发文章的四个地方，欢迎大家在朋友圈等地方分享，欢迎点“在看”。<br>我的个人博客地址：<a href="https://zwdnet.github.io/">https://zwdnet.github.io</a><br>我的知乎文章地址： <a target="_blank" rel="noopener" href="https://www.zhihu.com/people/zhao-you-min/posts">https://www.zhihu.com/people/zhao-you-min/posts</a><br>我的博客园博客地址： <a target="_blank" rel="noopener" href="https://www.cnblogs.com/zwdnet/">https://www.cnblogs.com/zwdnet/</a><br>我的微信个人订阅号：赵瑜敏的口腔医学学习园地</p>
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